   Chapter 15.3, Problem 7E

Chapter
Section
Textbook Problem

Evaluate the given integral by changing to polar coordinates.7. ∬ D x 2 y   d A , where D is the top half of the disk with center the origin and radius 5

To determine

To evaluate: The given integral by changing into the polar coordinates.

Explanation

Given:

The function, f(x,y)=x2y .

The region D is the upper semicircle centered at origin and radius 5.

Formula used:

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π , then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Calculation:

From the given region D, it is observed that the value of r varies from 0 to 5 and the value of θ varies from 0 to π .

Substitute x=rcosθ and y=rsinθ in the equation (1),

Dx2ydA=0π05(rcosθ)2(rsinθ)rdrdθ=0π05r4cos2θsinθdrdθ

Integrate the function with respect to θ and r by using the equation (2)

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